This can be the main finished survey of the mathematical lifetime of the mythical Paul Erd? s, probably the most flexible and prolific mathematicians of our time. For the 1st time, the entire major components of Erd? s' learn are coated in one undertaking. as a result of overwhelming reaction from the mathematical neighborhood, the undertaking now occupies over 900 pages, prepared into volumes.

Extra resources for Solvability and properties of solutions of nonlinear elliptic equations

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7. ,~ i A" (0) u. ]Lw, ]P-2. 9). 8. F o r N-> N3 the field r on SN(0, r) is l i n e a r l y homotopic to the field N = f A" (o) i=1 To compute the r o t a t i o n of the field X~ ) (u) we introduce the a u x i l i a r y o p e r a t o r A : W~,~(a) - - W ~ , ~ ( a ) defined by =IA'(O)u. Lvdx. 4 is applicable to the o p e r a t o r A if r 0 is defined by < r0u, v > = ! [ L - - A ' (0)1 u. Lvdx. 4 thus implies the following result. 7. Let conditions 1) and 2) be s a t i s f i e d , and suppose that z e r o is a nondegenerate c r i t i c a l point of the o p e r a t o r A defined by Eq.

We shall denote the rotation of Au on S by "y(Au, S). 9. In [99] it was shown that the concept of rotation introduced above can be extended to b r o a d e r c l a s s e s of o p e r a t o r s , and s o m e a s s u m p t i o n s on the space X can also be dropped. In p a r t i c u l a r , it is possible to drop the a s s u m p t i o n s r e g a r d i n g the s e p a r a b i l i t y and reflexivity of the s p a c e X. It is possible to define a s i n g l e valued r o t a t i o n of the v e c t o r field Au on S for a bounded, demicontinuous, pseudomonotone o p e r a t o r A if 0 ~ AS.